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Irreducibility of automorphic Galois representations of $G L (n)$ , $n$ at most $5$

Frank Calegari; Toby Gee — 2013

Annales de l’institut Fourier

Let $π$ be a regular, algebraic, essentially self-dual cuspidal automorphic representation of ${GL}_{n} (𝔸_{F})$ , where $F$ is a totally real field and $n$ is at most $5$ . We show that for all primes $l$ , the $l$ -adic Galois representations associated to $π$ are irreducible, and for all but finitely many primes $l$ , the mod $l$ Galois representations associated to $π$ are also irreducible. We also show that the Lie algebras of the Zariski closures of the $l$ -adic representations are independent of $l$ .

Local-global compatibility for $l = p$ , I

Thomas Barnet-Lamb; Toby Gee; David Geraghty; Richard Taylor — 2012

Annales de la faculté des sciences de Toulouse Mathématiques

We prove the compatibility of the local and global Langlands correspondences at places dividing $l$ for the $l$ -adic Galois representations associated to regular algebraic conjugate self-dual cuspidal automorphic representations of ${GL}_{n}$ over an imaginary CM field, under the assumption that the automorphic representations have Iwahori-fixed vectors at places dividing $l$ and have Shin-regular weight.

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Irreducibility of automorphic Galois representations of G L ( n ) , n at most 5

Local-global compatibility for l = p , I

Irreducibility of automorphic Galois representations of $G L (n)$ , $n$ at most $5$

Local-global compatibility for $l = p$ , I